AbstractA linearly ordered set A is said to shuffle into another linearly ordered set B if there is an order preserving surjection A —> B such that the preimage of each member of a cofinite subset of B has an arbitrary pre-defined finite cardinality. We show that every countable linearly ordered set shuffles into itself. This leads to consequences on transformations of subsets of the real numbers by order preserving maps.
AbstractWe use a new construction of an o-minimal structure, due to Lipshitz and Robinson, to answer a question of van den Dries regarding the relationship between arbitrary o-minimal expansions of real closed fields and structures over the real numbers. We write a first order sentence which is true in the Lipshitz-Robinson structure but fails in any possible interpretation over the field of real numbers.
AbstractContinuing earlier studies over the real numbers, we study the expressible set of a sequence A = (an)n≥1 of p-adic numbers, which we define to be the set EpA = {∑n≥1ancn: cn ∈ ℕ}. We show that in certain circumstances we can calculate the Haar measure of EpA exactly. It turns out that our results extend to sequences of matrices with p-adic entries, so this is the setting in which we work.
Abstract
In this paper a problem of approximating the real numbers by using the series of real numbers is considered. It is proven that if the given family of sequences of real numbers satisfies some conditions of set-theoretical nature, like being closed under initial subsequences and (additionally) possessing properties of adding and removing elements, then it automatically possesses some approximating properties, like, for example, reaching supremum of the set of sums of subseries.
Abstract
By rewriting the relation
1
+
2
=
3
{1+2=3}
as
1
2
+
2
2
=
3
2
{\sqrt{1}^{2}+\sqrt{2}^{2}=\sqrt{3}^{2}}
, a right triangle is looked at. Some
geometrical observations in connection with plane parqueting lead to
an inductive sequence of right triangles with
1
2
+
2
2
=
3
2
{\sqrt{1}^{2}+\sqrt{2}^{2}=\sqrt{3}^{2}}
as initial one converging to the
segment
[
0
,
1
]
{[0,1]}
of the real line. The sequence of their hypotenuses
forms a sequence of real numbers which initiates some beautiful
algebraic patterns. They are determined through some recurrence
relations which are proper for being evaluated with computer
algebra.
Abstract
We prove that when q is a power of 2 every complex irreducible representation of $\textrm{Sp}\big (2n, \mathbb{F}_{q}\big )$ may be defined over the real numbers, that is, all Frobenius–Schur indicators are 1. We also obtain a generating function for the sum of the degrees of the unipotent characters of $\textrm{Sp}\big(2n, \mathbb{F}_{q}\big )$, or of $\textrm{SO}\big(2n+1,\mathbb{F}_{q}\big )$, for any prime power q.
A question of van den Dries and a theorem of Lipshitz and Robinson; Not everything is standard
